Question - Optimizing Linear Programming with Constraints
Solution:
The image contains a linear programming problem that asks to minimize and maximize the objective function P = 25x + 6y given the constraints:1. x + y ≤ 232. 4x + 5y ≤ 433. x ≥ 54. y ≥ 0Firstly, we want to find the feasible region defined by the constraints and then locate the corner points since the minimum and maximum values for P will occur at the corners of this region due to the nature of linear programming problems.Let's find the intersection points of the lines given by the constraints:From the first constraint, if x = 5 (the minimum value for x), we can find the maximum y:5 + y ≤ 23 → y = 18From the second constraint, if x = 5, we can find y:4(5) + 5y ≤ 43 → 20 + 5y ≤ 43 → 5y ≤ 23 → y = 4.6 (but since y needs to be an integer, y = 4)Now let's see where the first and second constraints intersect:x + y = 23 and 4x + 5y = 43 can be solved as a system of equations. However, rather than solving it algebraically, it might be easier to graph these constraints and visually find the intersection points, especially since we know they have to be integers due to the contexts of these problems.Graphing these lines, along with x ≥ 5 and y ≥ 0, we notice that the possible intersection points (i.e., the corner points of the feasible region) are:- (5, 18) from x = 5 and x + y = 23- (5, 4) from x = 5 and 4x + 5y = 43- The intersection of x + y = 23 and 4x + 5y = 43, which we will calculate.Solving the system of equations:x + y = 234x + 5y = 43Multiply the first equation by 4:4x + 4y = 924x + 5y = 43Subtract the second equation from the multiplied first one:4y - 5y = 92 - 43-y = 49y = -49, which is not possible since y ≥ 0, so this intersection is not within the feasible region.Thus, the corner points of the feasible region are (5, 18) and (5, 4) because the intersection of the two constraints falls outside the feasible region. Let's now find the values of P at these corner points:For (5, 18):P = 25(5) + 6(18) = 125 + 108 = 233For (5, 4):P = 25(5) + 6(4) = 125 + 24 = 149Now, comparing the values, we see that the minimum value of P is 149 at the point (5, 4).Answering the questions:- What is the minimum value of P? A: 149- What are the coordinates of the corner point where the minimum value of P occurs?A: (5, 4)
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com